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Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue

In this page, a simple overview of the divisibility test for `9` is provided.



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Which of the following are multiples of `9`?

  • `9,18,27,36, cdots`
  • `9,18,27,36, cdots`
  • `10,20,30,40,cdots`

The answer is "`9,18,27,36, cdots`"

Consider the multiples of `9` given as `9,18,27,36, cdots`. Is there any similarity observed in the multiples?

  • all are ending in `9`
  • sum of all digits is a multiple of `9`
  • sum of all digits is a multiple of `9`

The answer is "sum of all digits is a multiple of `9`".

This is explained as follows
`18 -> 1+8=9` sum is `9`
`27 -> 2+7=9` sum is `9`
`36 -> 3+6=9` sum is `9`
`189 -> 1+8+9=18` sum is multiple of `9`

This is true for all multiples of `9`.

Consider the numbers `12` and `81`. Which one is divisible by `9`?

Check this by long division method.

  • `81`
  • `81`
  • `12`

The answer is "`81`"

To identify the multiplies of `9`, the sum of all digits of the number is checked for divisibility by `9`.

Explanation for the curious mind.

Consider divisibility test `42` by `9`

To simplify the divisibility test, let us subtract a multiple of the divisor `9`.
Seeing the `10`s place value `4`, we choose the multiple `9xx4` to subtract from the number.

As per the property of simplification by subtraction, the divisibility test of `42` is simplified into divisibility test of `42-36 = 40-36+2 = 4+2`.

This explains the divisibility test for `9`.

Test for Divisibility by `9` : If the sum of all the digits is divisible by `9`, then the number is divisible by `9`

Is `2008` divisible by `9`?

  • Yes
  • No
  • No

The answer is "No". Checking the sum of digits `2+0+0+8=10`, it is concluded that the number is not divisible by `9`.

Is `927` divisible by `9`?

  • Yes
  • Yes
  • No

The answer is "Yes". Checking the sum of digits `9+2+7=18`, it is concluded that the number is divisible by `9`.

                            
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